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On partitions of discrete boxes

Noga Alon, Tom Bohman, Ron Holzman, Daniel J. Kleitman
2002 Discrete Mathematics  
We prove that any partition of an n-dimensional discrete box into nontrivial sub-boxes must consist of at least 2 n sub-boxes, and consider some extensions of this theorem.  ...  We are also grateful to Jeff Kahn for suggesting the point of view taken in subsection 2.4.  ...  Theorem 1 Let A be an n-dimensional discrete box, and let {B 1 , B 2 , . . . , B m } be a partition of A into nontrivial sub-boxes. Then m ≥ 2 n . Proof.  ... 
doi:10.1016/s0012-365x(02)00428-4 fatcat:gsi3xpnyhnfthcf5lqhqcn5spq

On Partitions of Two-Dimensional Discrete Boxes [article]

Eyal Ackerman, Rom Pinchasi
2018 arXiv   pre-print
Let A and B be finite sets and consider a partition of the discrete box A × B into sub-boxes of the form A' × B' where A' ⊂ A and B' ⊂ B.  ...  We show that a partition of A × B that has the (k, ℓ)-piercing property must consist of at least (k-1)+(ℓ-1)+ 2√((k-1)(ℓ-1)) sub-boxes.  ...  Acknowledgments We thank Ron Holzman for pointing out a mistake in an earlier version of this paper.  ... 
arXiv:1812.08396v1 fatcat:eblbosvmcvdz3j765y5g6kmgym

On the number of minimal partitions of a box into boxes

Andrzej P. Kisielewicz, Krzysztof Przesławski
2006 Discrete Mathematics  
The number of minimal partitions of a box into proper boxes is examined.  ...  At the present stage of investigations this problem seems to be out of reach, as we know little on the structure of the minimal partitions.  ...  to F satisfies one of the following two relations: Lemma 6.  ... 
doi:10.1016/j.disc.2005.10.027 fatcat:7bseg7ttpzccvjbjr6paxqmjcy

Performing large full-wave simulations by means of a parallel MLFMA implementation

B. Michiels, J. Fostier, I. Bogaert, D. De Zutter
2013 2013 IEEE Antennas and Propagation Society International Symposium (APSURSI)  
The data structures of the MLFMAtree are partitioned according to the so-called hierarchical partitioning scheme, while the radiation patterns are partitioned in a blockwise way.  ...  To test the implementation of the algorithm, a full-wave simulation of a canonical example with more than 50 millions of unknowns has been performed.  ...  This method discretizes the scattering problem into a set of N linear equations, with N the number of unknowns on the discretized surface(s) of the scatterer(s).  ... 
doi:10.1109/aps.2013.6711598 fatcat:uhznxf3oznflvm6kitx2ec7eie

Hierarchical parallelization of the multilevel fast multipole algorithm (MLFMA)

L. Gurel, Ozgur Ergul
2013 Proceedings of the IEEE  
This paper presents the hierarchical partitioning strategy, which provides a very efficient parallelization of MLFMA on distributed-memory architectures.  ...  We discuss the advantages of the hierarchical strategy over previous approaches and demonstrate the improved efficiency on scattering problems discretized with millions of unknowns.  ...  One of the major advances in the parallelization of MLFMA is the development of the hierarchical strategy, which is based on the simultaneous partitioning of boxes and field samples among processors at  ... 
doi:10.1109/jproc.2012.2222331 fatcat:wvgih5zlmbdlhenmccrvu4mywa

Range-Box Plotting Relating to Discrete Distribution

Mohd Bakri Adam, Babaginda Ibrahim Babura, Kathiresan Gopal
The new modication of box plot is not based on the asymmetrical shape of distribution but more on the spread of the data and partitioning data into range measure.  ...  The existing box plot is constructed using ve values of statistics calculated from either the discrete or continous data.  ...  Table 3 shows that the standard classical box plot will not look at the partitional of data, but really on the existance of outliers.  ... 
doi:10.11113/matematika.v34.n2.935 fatcat:wdmyh6jjtbdajlpim2rgzswmu4

Simulating Microbial Community Patterning Using Biocellion [chapter]

Seunghwa Kang, Simon Kahan, Babak Momeni
2014 Msphere  
Biocellion further accelerates large scale, high resolution simulations using cluster computers by partitioning the work to run on multiple compute nodes.  ...  Biocellion is a highperformance software framework for accelerating discrete agent-based simulation of biological systems with millions to trillions of cells.  ...  The original yeast patterning model [5] partitions the simulation domain on top of the agarose cylinder into a set of fixed size cubic boxes (a box width is 5 μm), and a cell takes a single box.  ... 
doi:10.1007/978-1-4939-0554-6_16 pmid:24838890 fatcat:kmmuib5y6van5d4cgrgckh3efq

The maximum discrete surface-to-volume ratio of space-filling curve partitions [article]

Maximilien Gadouleau, Tobias Weinzierl
2021 arXiv   pre-print
This framework thus has the potential to guide the design of better load balancing algorithms on the long term.  ...  We investigate the surface-to-volume ratio of space-filling curve partitions for adaptive grids and derive the maximum surface-to-volume ratio as a function of the number of cells in the partition.  ...  A fundamental goal in this project is the effective parallelisation of a dynamically adaptive mesh refinement solver for wave questions which uses the Peano space-filling curve.  ... 
arXiv:2106.12856v1 fatcat:vef5pujombhebbmjbmt4fid46i

Biocellion: accelerating computer simulation of multicellular biological system models

S. Kang, S. Kahan, J. McDermott, N. Flann, I. Shmulevich
2014 Bioinformatics  
Availability and implementation: Biocellion runs on x86 compatible systems with the 64 bit Linux operating system and is freely available for academic use.  ...  Discrete agentbased modeling (in combination with high-resolution grids to model the extracellular environment) is a popular approach for building biological system models.  ...  The extracellular space is partitioned into a set of cubical grid boxes.  ... 
doi:10.1093/bioinformatics/btu498 pmid:25064572 pmcid:PMC4609016 fatcat:j4zdhyz7vfcqzn6az6h3zayfse

Enabling Space Filling Curves Parallel Mesh Partitioning In Alya

R. Borrell
2017 Zenodo  
In this paper we focus on the mesh partitioning, presenting the work carried out in the context of a PRACE Preparatory Access Project to enable a Space Filling Curve (SFC) based partitioner in the computational  ...  In particular, we have run our tests on the MareNostrum III supercomputer of the Barcelona Supercomputing Center.  ...  The sequential and parallel executions of the algorithm produce an almost identical partition if the grid used to discretize the bounding box is the same.  ... 
doi:10.5281/zenodo.826103 fatcat:zsub22ti7jgffg2gztcziaguwu

Optimistic planning for continuous-action deterministic systems

Lucian Busoniu, Alexander Daniels, Remi Munos, Robert Babuska
2013 2013 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning (ADPRL)  
Experiments on three problems show SOOP reliably ranks among the best algorithms, fully dominating competing methods when the problem requires both long horizons and fine discretization.  ...  SOOP can be used parameter-free at the cost of more model calls, but we also propose a more practical variant tuned by a parameter α, which balances finer discretization with longer planning horizons.  ...  The hierarchical partition splits at each depth d the considered box along dimension k = d, into M subboxes: one for each discrete action. So, for OPD the depth in the tree is equal to the time step.  ... 
doi:10.1109/adprl.2013.6614991 dblp:conf/adprl/BusoniuDMB13 fatcat:em4v6xs5zjcwtbs7oc6hub4wsq

FAUST $^{\mathsf 2}$ : Formal Abstractions of Uncountable-STate STochastic Processes [chapter]

Sadegh Esmaeil Zadeh Soudjani, Caspar Gevaerts, Alessandro Abate
2015 Lecture Notes in Computer Science  
FAUST 2 allows refining the outcomes of the verification procedures over the concrete dtMP in view of the quantified and tunable error, which depends on the dtMP dynamics and on the given formula.  ...  FAUST 2 is a software tool that generates formal abstractions of (possibly non-deterministic) discrete-time Markov processes (dtMP) defined over uncountable (continuous) state spaces.  ...  Left: obtained uniform partition of the safe set, along with optimal safety probability for each partition set (color bar on the right).  ... 
doi:10.1007/978-3-662-46681-0_23 fatcat:qqltkzrnkvhdvkq6m7ld7ur6dq

Sylvester's bijection between strict and odd partitions

Alain Lascoux
2004 Discrete Mathematics  
We show that Sylvester's bijection between strict partitions and odd ones can be obtained by an appropriate coding of partitions.  ...  The contents of the boxes may be considered as materializing a ribbon made of two parts, the top one being described by its vertical steps, the bottom one, by its horizontal steps, each number being the  ...  As a by-product, one obtains a statistics on odd partitions which generalizes the restriction of an identity of Bousquet-Melou and Eriksson to strict partitions (cf. [3] ).  ... 
doi:10.1016/j.disc.2002.02.001 fatcat:gkjbf2gs7rhedlvfbcgdnzj3ry

Fast and accurate solutions of extremely large scattering problems involving three-dimensional canonical and complicated objects

Ozgur Ergul, Levent Gurel
2009 2009 Computational Electromagnetics International Workshop  
We present fast and accurate solutions of extremely large scattering problems involving three-dimensional metallic objects discretized with hundreds of millions of unknowns.  ...  Various examples involving canonical and complicated objects are presented in order to demonstrate the feasibility of accurately solving large-scale problems on relatively inexpensive computing platforms  ...  At this frequency, the size of the box corresponds to 875λ, and its discretization with the RWG functions on λ/10 triangles leads to matrix equations involving 174,489,600 unknowns.  ... 
doi:10.1109/cem.2009.5228112 fatcat:4iiqsuesvrgofkdbpn74g4vodu

How many is different? Answer from ideal Bose gas

Jeong-Hyuck Park
2014 Journal of Physics, Conference Series  
We show that the isobar of the ideal gas zigzags on the temperature-volume plane featuring a 'boiling-like' discrete phase transition, provided the number of particles is equal to or greater than a particular  ...  We revisit with care the ideal Bose gas confined in a cubic box which is discussed in most statistical physics textbooks.  ...  This work was supported by the Sogang University Research Grant of 201310030.01.  ... 
doi:10.1088/1742-6596/490/1/012018 fatcat:jqxnno5kz5dztjvwjnq4t7amvm
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